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State Board · Class 9 · Mathematics Part 2

Lesson 3: Triangles

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Theorem of remote interior angles

From daily life

Stand at the corner of a triangular garden and turn your body from the line of one side to the line of the next. That turn is the exterior angle, and it equals the two far-away corner angles added together.

Simple definition

Theorem of remote interior angles (दूरच्या अंतर्कोनांचे प्रमेय): an exterior angle of a triangle equals the sum of its two remote interior angles.

Rule 1: The theorem — In ΔPQR, extend QR to S. Then ∠PRS = ∠PQR + ∠QPR. Proof: ∠PQR + ∠QPR + ∠PRQ = 180° and ∠PRQ + ∠PRS = 180° (linear pair). Removing ∠PRQ from both gives the result. So an exterior angle is greater than each remote interior angle.

A triangle PQR with side QR extended to S; the exterior angle at R is 110 degrees, the sum of the 50 and 60 degree anglesPQRS50°60°110°Exterior ∠PRS = 50° + 60° = 110°

In the figure : In the figure the exterior angle equals the sum of the two remote interior angles (for example 55° + 65° = 120°).

Live figure

Theorem of remote interior angles

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  • Easy : Easy: ∠A = 55° and ∠B = 65°, so the exterior angle at C is 120°.
  • Medium : Medium: Exterior angle 130°, one remote angle 70°, so the other is 60°.
  • Exam-type : Exam: Exterior angle (3x + 10)° and remote angles 2x and 50°: 3x + 10 = 2x + 50, so x = 40 and the exterior angle is 130°.

Rule 2: Angle ratios and exterior angle sum — Angles in the ratio 2 : 3 : 4: 9x = 180°, x = 20°, so 40°, 60°, 80°. Angles x, 2x, 3x give 6x = 180°, so 30°, 60°, 90°. If one exterior angle is taken at each vertex (sides extended in the same direction), the three exterior angles add up to 360°.

  • Easy : Easy: Exterior angles 120° and 130° mean the third is 360° − 250° = 110°.
  • Medium : Medium: Angles 3x, 4x, 5x: 12x = 180°, x = 15°, so 45°, 60°, 75°.
  • Exam-type : Exam: In an equilateral triangle each exterior angle is 180° − 60° = 120°, and 3 × 120° = 360°.

Rule 3: Angle bisectors meeting at P — If the bisectors of ∠B and ∠C of ΔABC meet at P, then ∠BPC = 90° + 12∠BAC. For ∠A = 60°: ∠BPC = 90° + 30° = 120°. Check: half of (∠B + ∠C) = half of 120° = 60°, and 180° − 60° = 120°.

  • Easy : Easy: ∠A = 40° gives ∠BPC = 90° + 20° = 110°.
  • Medium : Medium: ∠A = 80° gives ∠BPC = 90° + 40° = 130°.
  • Exam-type : Exam: If ∠BPC = 125°, then 12∠A = 35°, so ∠A = 70°.
Remember

Exterior angle = sum of the two remote interior angles. Triangle angles add to 180°.

Common mistake

Adding the adjacent interior angle to find the exterior angle. Use only the two remote ones.

Shortcut

Memory clue: "बाह्य = दूरचे दोन".

Spot the difference

FactValue
Angle sum180°
Exterior angleSum of 2 remote angles
Three exterior angles360°

Practice questions

Easy
  1. 1State the theorem of remote interior angles.
  2. 2∠A = 45°, ∠B = 50°. Exterior angle at C?
  3. 3Exterior angle 110°, one remote angle 40°. The other?
Medium
  1. 4Sum of three exterior angles (one at each vertex)?
  2. 5Angles in the ratio 1 : 1 : 2. Find them.
  3. 6∠A = 100°. Find ∠BPC for the angle bisectors of B and C.
Challenging
  1. 7Exterior angle (2x + 30)°, remote angles x° and 60°. Find x.
  2. 8Why is an exterior angle bigger than each remote angle?
Show answers (try them yourself first)
  1. 1The exterior angle equals the sum of the two remote interior angles.
  2. 295°.
  3. 370°.
  4. 4360°.
  5. 545°, 45°, 90°.
  6. 690° + 50° = 140°.
  7. 72x + 30 = x + 60, x = 30.
  8. 8It equals their sum, and a sum of two positive angles is more than either.

Congruence of triangles

From daily life

Two identical rangoli stencils fit exactly on each other. Triangles that fit exactly are congruent, so all their matching sides and angles are equal.

Simple definition

Triangles are congruent (एकरूप), ΔABC ≅ ΔPQR, if all three pairs of sides and all three pairs of angles match. The tests need fewer facts.

Rule 1: Four congruence tests — S-S-S: three sides equal. S-A-S: two sides and the included angle equal. A-S-A: two angles and the included side equal. Hypotenuse-side: both right-angled, equal hypotenuses and one more pair of equal sides. If ΔABC ≅ ΔPQR, then ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and AB = PQ, BC = QR, CA = RP.

The four tests of congruence of triangles; equal marks show equal partsA-S-Atwo angles and the included sideS-A-Stwo sides and the included angleS-S-Sthree sidesHypotenuse-sideright angle, hypotenuse and one side

In the figure : In the figure the four congruence tests are shown with marks for equal parts.

  • Easy : Easy: AB = PQ = 6, BC = QR = 8, CA = RP = 7: congruent by S-S-S.
  • Medium : Medium: AB = PQ = 5, AC = PR = 7 and ∠A = ∠P = 40°: congruent by S-A-S.
  • Exam-type : Exam: ∠B = ∠Q = 60°, BC = QR = 4, ∠C = ∠R = 70°: congruent by A-S-A.

Rule 2: Using congruence — Corresponding sides of congruent triangles are c.s.c.t.; corresponding angles are c.a.c.t. In ΔABD and ΔCBD, AB = CB, AD = CD and BD is common, so ΔABD ≅ ΔCBD (S-S-S), and then ∠A = ∠C (c.a.c.t.). Equal angles alone (A-A-A) do not prove congruence: a 1 cm and a 2 cm equilateral triangle have the same angles.

  • Easy : Easy: A common side counts as one pair of equal sides.
  • Medium : Medium: Right triangles with hypotenuse 13 and one side 5 each are congruent (hypotenuse-side).
  • Exam-type : Exam: Why is S-S-A not a test? The angle must be between the two sides (included).
Remember

Write the vertices in matching order. Check the tests: SSS, SAS, ASA, hypotenuse-side.

Common mistake

Using A-A-A or S-S-A. These do not prove congruence.

Shortcut

Memory clue: "SSS, SAS, ASA, कर्ण-बाजू".

Spot the difference

TestNeeds
S-S-S3 sides
S-A-S2 sides + included angle
A-S-A2 angles + included side
Hypotenuse-sideRight angle, hypotenuse, 1 side

Practice questions

Easy
  1. 1When are two triangles congruent?
  2. 2Name the four congruence tests.
  3. 3What does c.a.c.t. stand for?
Medium
  1. 4AB = PQ, BC = QR, CA = RP. Which test?
  2. 5Two sides and a non-included angle equal. Congruent?
  3. 6ΔABC ≅ ΔPQR. If ∠B = 50°, what is ∠Q?
Challenging
  1. 7In ΔABD and ΔCBD, AB = CB, AD = CD. Prove ∠A = ∠C.
  2. 8Why is A-A-A not a congruence test?
Show answers (try them yourself first)
  1. 1When one fits exactly on the other.
  2. 2S-S-S, S-A-S, A-S-A, hypotenuse-side.
  3. 3Corresponding angles of congruent triangles.
  4. 4S-S-S.
  5. 5Not necessarily.
  6. 650°.
  7. 7BD is common, so the triangles are congruent by S-S-S; hence ∠A = ∠C by c.a.c.t.
  8. 8Triangles of different sizes can have the same angles.

Isosceles triangle and special right triangles

From daily life

A kite's string makes equal angles with the two equal sticks of its frame. A folded paper triangle shows the same: when two sides match, so do the angles opposite to them.

Simple definition

Isosceles triangle theorem (समद्विभुज त्रिकोण प्रमेय): equal sides have equal opposite angles, and the converse is also true.

Rule 1: Isosceles triangle — If AB = AC then ∠B = ∠C (draw the bisector of ∠BAC meeting BC at D; ΔABD ≅ ΔACD by S-A-S). Converse: if ∠B = ∠C then AB = AC. If all sides are equal, all angles are equal (each 60°), and conversely.

An isosceles triangle ABC with equal sides AB and AC and equal base anglesxxABCDAB = AC ⇒ ∠B = ∠C

In the figure : In the figure equal sides have equal opposite angles.

  • Easy : Easy: AB = AC and ∠B = 65° give ∠C = 65° and ∠A = 50°.
  • Medium : Medium: Apex angle 80°: each base angle is (180° − 80°) ÷ 2 = 50°.
  • Exam-type : Exam: ∠B = ∠C = 55° means AB = AC, and ∠A = 70°.

Rule 2: 30°-60°-90° and 45°-45°-90° triangles — 30°-60°-90°: the side opposite 30° is half the hypotenuse, and the side opposite 60° is √32 × hypotenuse. Hypotenuse 12: sides 6 and 6√3. 45°-45°-90°: each side containing the right angle is 1√2 × hypotenuse. Hypotenuse 8: each side is 4√2.

The 30-60-90 triangle with sides 1, root 3 and 2 and the 45-45-90 triangle with sides 1, 1 and root 21√3230°60°30° – 60° – 90°11√245°45°45° – 45° – 90°Side ratios of the two special right triangles

In the figure : In the figure side ratios of the 30°-60°-90° triangle (1 : √3 : 2) and the 45°-45°-90° triangle (1 : 1 : √2).

  • Easy : Easy: In a 30°-60°-90° triangle, a 4 cm short side means hypotenuse 8 cm.
  • Medium : Medium: A 45°-45°-90° triangle with sides 5 cm has hypotenuse 5√2 cm.
  • Exam-type : Exam: A 30°-60°-90° triangle with hypotenuse 20 cm has sides 10 cm and 10√3 cm.
Remember

Equal sides, equal opposite angles. 30°-60°-90°: 1 : √3 : 2. 45°-45°-90°: 1 : 1 : √2.

Common mistake

Putting √3 on the wrong side. √3 goes with 60°, 1 goes with 30°.

Shortcut

Memory clue: "30° → निम्मा; 60° → √3 × निम्मा; 45° → कर्ण ÷ √2".

Spot the difference

TriangleSides
IsoscelesEqual sides, equal opposite angles
30°-60°-90°1 : √3 : 2
45°-45°-90°1 : 1 : √2

Practice questions

Easy
  1. 1State the isosceles triangle theorem.
  2. 2Each angle of an equilateral triangle?
  3. 3AB = AC, ∠A = 100°. Find ∠B.
Medium
  1. 4In a 30°-60°-90° triangle the hypotenuse is 10. Short side?
  2. 5In a 45°-45°-90° triangle the sides are 7. Hypotenuse?
  3. 645°-45°-90° triangle with hypotenuse 10 cm: sides?
Challenging
  1. 7∠B = ∠C = (2x + 5)° and ∠A = 50°. Find x.
  2. 8Which two ideas prove the isosceles triangle theorem?
Show answers (try them yourself first)
  1. 1Equal sides have equal opposite angles.
  2. 260°.
  3. 340°.
  4. 45.
  5. 57√2.
  6. 610 ÷ √2 = 5√2 cm each.
  7. 74x + 10 = 130, x = 30.
  8. 8The angle bisector and S-A-S congruence.

Medians, perpendicular bisector and angle bisector

From daily life

Balance a cardboard triangle on a pencil tip. It balances at one special point, where the three lines drawn from corners to the middle of opposite sides meet.

Simple definition

A median (मध्यगा) joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid (गुरुत्वमध्य).

Rule 1: Median and centroid — The centroid G divides each median in the ratio 2 : 1 (longer part towards the vertex). If median AD = 9 cm, then AG = 6 cm and GD = 3 cm. If GD = 4 cm then AG = 8 cm. In a right-angled triangle the median on the hypotenuse is half the hypotenuse: hypotenuse 18 gives median 9.

The three medians of a triangle meet at the centroid G, which divides each median in the ratio 2 to 1GABCDEFCentroid G: AG : GD = 2 : 1 (if GD = 3 then AG = 6)

In the figure : In the figure the medians meet at the centroid, which divides each in the ratio 2 : 1.

  • Easy : Easy: Median 12 cm gives AG = 8 cm and GD = 4 cm.
  • Medium : Medium: GD = 2.5 cm gives AG = 5 cm and the whole median 7.5 cm.
  • Exam-type : Exam: Right triangle with hypotenuse 26: median on the hypotenuse is 13.

Rule 2: Perpendicular bisector theorem — Every point on the perpendicular bisector of a segment is equidistant from its end points (S-A-S). Any point equidistant from the end points of a segment lies on its perpendicular bisector (S-S-S). If PA = 7 for P on the bisector of seg AB, then PB = 7.

  • Easy : Easy: P on the bisector with PA = 5 gives PB = 5.
  • Medium : Medium: QA = QB means Q lies on the perpendicular bisector.
  • Exam-type : Exam: The bisector of seg AB with AB = 10 meets it at M with AM = 5.

Rule 3: Angle bisector theorem — Every point on the bisector of an angle is equidistant from the sides of the angle. Any point inside an angle equidistant from its sides lies on its bisector. If a point on the bisector is 4 cm from one arm, it is 4 cm from the other.

Live figure

Medians, perpendicular bisector and angle bisector

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  • Easy : Easy: Distance 3 cm from one arm gives 3 cm from the other.
  • Medium : Medium: A point 6 cm from both arms lies on the bisector.
  • Exam-type : Exam: The three angle bisectors of a triangle meet at a point equidistant from all three sides.
Remember

Centroid: 2 : 1 on each median. Right triangle: median on hypotenuse = half hypotenuse.

Common mistake

Dividing the median 1 : 2 from the vertex. The longer part is next to the vertex.

Shortcut

Memory clue: "गुरुत्वमध्य : शिरोबिंदू 2, बाजू 1".

Spot the difference

IdeaFact
MedianVertex to midpoint
CentroidDivides median 2 : 1
Perpendicular bisectorEqual distance from end points
Angle bisectorEqual distance from arms

Practice questions

Easy
  1. 1What is a median?
  2. 2How many medians has a triangle?
  3. 3Where do they meet?
Medium
  1. 4In what ratio does G divide a median?
  2. 5GD = 3 cm. Find AG and AD.
  3. 6Hypotenuse = 30 cm. Median on it?
Challenging
  1. 7State the perpendicular bisector theorem.
  2. 8P is on the bisector of ∠XOY and is 5 cm from OX. Distance from OY?
Show answers (try them yourself first)
  1. 1A segment from a vertex to the midpoint of the opposite side.
  2. 23.
  3. 3At the centroid.
  4. 42 : 1.
  5. 5AG = 6 cm, AD = 9 cm.
  6. 615 cm.
  7. 7Points on it are equidistant from the segment's end points, and conversely.
  8. 85 cm.

Inequalities in a triangle and similar triangles

From daily life

A longer stick always sits opposite a wider opening. And two short sticks of 2 cm and 3 cm can never reach round a 9 cm gap. These ideas give the inequalities of a triangle.

Simple definition

The greater side is opposite the greater angle, and any two sides add up to more than the third. Shapes with equal angles and proportional sides are similar (समरूप).

Rule 1: Inequalities — If two sides are unequal, the angle opposite the greater side is greater; if two angles are unequal, the side opposite the greater angle is greater. The sum of any two sides is greater than the third. Sides 5 cm and 9 cm need a third side between 4 cm and 14 cm. Sides 4, 5, 10 are impossible since 4 + 5 = 9 < 10.

A triangle with sides 5, 7 and 8 showing that the greater side is opposite the greater angleABCBC = 7AC = 5AB = 8Biggest side AB is opposite biggest angle C; 5 + 7 = 12 > 8Bigger side ↔ bigger opposite angle

In the figure : In the figure the greater side is opposite the greater angle, and the sum of two sides is more than the third.

  • Easy : Easy: ∠A = 70° and ∠B = 50°, so BC (opposite A) is longer than AC (opposite B).
  • Medium : Medium: Can 6, 7, 12 be sides? 6 + 7 = 13 > 12, yes.
  • Exam-type : Exam: Can 7, 10, 17 be sides? 7 + 10 = 17, not greater than 17, so no.

Rule 2: Similar triangles — ΔABC ~ ΔPQR if ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and ABPQ = BCQR = ACPR. If the angles of two triangles are equal, the triangles are similar and corresponding sides are proportional. Example: ΔXYZ ~ ΔPQR, XY = 3, YZ = 5, XZ = 6, PQ = 9. Ratio = 9 ÷ 3 = 3, so QR = 15 and PR = 18.

Two similar triangles with the same angles, the bigger one with sides twice as longXYZPQR4 : 6 : 58 : 12 : 10Same angles, same shape: ratio of sides = 2XY : PQ = 4 : 8 = YZ : QR = XZ : PR = 1 : 2

In the figure : In the figure similar triangles have equal angles and sides in the same ratio.

  • Easy : Easy: Ratio 2: a side 4 becomes 8.
  • Medium : Medium: Sides 2, 3, 4 and 6, 9, 12 are similar (ratio 3).
  • Exam-type : Exam: Perimeters of similar triangles are in the same ratio: 12 and 36 for ratio 3.
Remember

Bigger side faces bigger angle. Two sides together beat the third.

Common mistake

Checking the wrong pair of sides. Compare the two shorter sides with the longest.

Shortcut

Memory clue: "दोन लहान मिळून मोठ्यापेक्षा मोठ्या".

Spot the difference

IdeaRule
Side and angleGreater side, greater opposite angle
Triangle ruleTwo sides > third
SimilarEqual angles, proportional sides

Practice questions

Easy
  1. 1Which side is opposite the greater angle?
  2. 2State the triangle inequality.
  3. 3Can 3, 4, 8 form a triangle?
Medium
  1. 4Can 5, 6, 10 form a triangle?
  2. 5What does ΔABC ~ ΔPQR mean?
  3. 6ΔABC ~ ΔPQR with ratio 2 and AB = 5. Find PQ.
Challenging
  1. 7Two sides of a triangle are 6 cm and 8 cm. Between what values is the third side?
  2. 8ΔABC ~ ΔPQR, AB = 4, PQ = 6, BC = 10. Find QR.
Show answers (try them yourself first)
  1. 1The greater side.
  2. 2Any two sides add up to more than the third.
  3. 3No, 3 + 4 = 7 < 8.
  4. 4Yes, 5 + 6 = 11 > 10.
  5. 5Equal angles and proportional sides.
  6. 610.
  7. 7Between 2 cm and 14 cm.
  8. 8Ratio 1.5, so QR = 15.

Quick revision

  • An exterior angle of a triangle equals the sum of its remote interior angles; the congruence tests are A-S-A, S-A-S, S-S-S and the hypotenuse side test.
  • Equal sides have equal opposite angles; in a 30°-60°-90° triangle the shortest side is half the hypotenuse; the median on the hypotenuse is half the hypotenuse.
  • The greater side is opposite the greater angle, and any two sides of a triangle add up to more than the third.